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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 06}
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%\date{2021 年 2 月 28 日}
\date{May 11, 2021}

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%\subsection{Homework 06}
%E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\item [E6.1.1.] A pure birth process starting from $X(0) = 0$ has birth parameters $\lambda_0 = 1$, $\lambda_1 = 3$, $\lambda_2 = 2$, and $\lambda_3 = 5$. Determine $P_n(t)$ for $n = 0,1,2,3$.


\item [E6.1.2.] A pure birth process starting from $X(0) = 0$ has birth parameters $\lambda_0 = 1$, $\lambda_1 =3$, $\lambda_2 =2$, and $\lambda_3 =5$. Let $W_3$ be the random time that it takes the process to reach state 3. 
\begin{enumerate}
\item  Write $W_3$ as a sum of sojourn times and thereby deduce that the mean time is $$\mathbb{E}(W_3)=11/6.$$
\item  Determine the mean of $W_1+W_2+W_3$. 
\item  What is the variance of $W_3$?
\end{enumerate}


\item [P6.1.1.] Let $X(t)$ be a Yule process that is observed at a random time $U$, where $U$ is uniformly distributed over $[0,1)$. Show that $\mathbb{P}\{X(U) = k\} = p^k/(\beta k)$ for $k = 1,2,\cdots$, with $p = 1-e^{-\beta}$.


\item [P6.1.2.] A Yule process with immigration has birth parameters $\lambda_k = \alpha + k\beta$ for $k = 0,1,2,\cdots$. 
Here, $\alpha$ represents the rate of immigration into the population, and $\beta$ represents the individual birth rate. 
Supposing that $X(0) = 0$, determine $P_n(t)$ for $n = 0,1,2,\cdots$.





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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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